Pavel Kurasov

Pavel Kurasov
Stockholm University | SU · Department of Mathematics

About

169
Publications
13,206
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
3,055
Citations
Additional affiliations
September 2010 - present
Stockholm University
Position
  • Professor (Full)

Publications

Publications (169)
Preprint
Quantum graphs have attracted attention from mathematicians for some time. A quantum graph is defined by having a Laplacian on each edge of a metric graph and imposing boundary conditions at the vertices to get an eigenvalue problem. A problem studying such quantum graphs is that the spectrum is timeconsuming to compute by hand and the inverse prob...
Preprint
We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoid...
Chapter
A relationship between the Euler characteristic of a quantum graph and its spectrum is a very new subject of the theoretical and experimental investigations. The Euler characteristic \(\chi =|V|-|E|\), where |V| and |E| are the numbers of vertices and edges of a graph, determines the number \(\beta \) of independent cycles in it. The most important...
Preprint
It is proven following [18[ that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not equal to zero in any non-Dirichlet vertex.
Preprint
A new class of isospectral graphs is presented. These graphs are isospectral with respect to both the normalised Laplacian on the discrete graph and the standard differential Laplacian on the corresponding metric graph. The new class of graphs is obtained by gluing together subgraphs with the Steklov maps possessing special properties. It turns out...
Article
Full-text available
The Euler characteristic i.e., the difference between the number of vertices | V | and edges | E | is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized E...
Article
Full-text available
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...
Article
Full-text available
Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have beco...
Article
Full-text available
The spectra of n -Laplacian operators $$(-\Delta )^n$$ ( - Δ ) n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved us...
Article
Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area.
Preprint
The Euler characteristic $\chi =|V|-|E|$ and the total length $\mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $\beta$ of independent cycles in a graph while the total length det...
Chapter
(Kronshtadt, Russia, 27 July 1936 – Auckland, New Zealand, 30 January 2016)
Chapter
Pavlov’s contribution to science is not limited to his publications, he used to say that papers should be written for political reasons. Nevertheless, most of Pavlov’s ideas are reflected in his publications showing us different facets of his scientific personality.
Article
The Euler characteristic χ=|V|−|E| and the total length L are the most important topological and geometrical characteristics of a metric graph. Here |V| and |E| denote the number of vertices and edges of a graph. The Euler characteristic determines the number β of independent cycles in a graph while the total length determines the asymptotic behavi...
Preprint
Full-text available
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...
Preprint
Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area.
Article
Full-text available
The theory of almost periodic functions is used to investigate spectral properties of Schrödinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schrödinger operators may have asymptotically close spectra if and only if the corresponding reference Laplacians are isospectral. Our result implies that a Schrö...
Book
Boris Pavlov (1936-2016), to whom this volume is dedicated, was a prominent specialist in analysis, operator theory, and mathematical physics. As one of the most influential members of the St. Petersburg Mathematical School, he was one of the founders of the Leningrad School of Non-self-adjoint Operators. This volume collects research papers orig...
Book
This book contains contributions from the participants of the research group hosted by the ZiF - Center for Interdisciplinary Research at the University of Bielefeld during the period 2013-2017 as well as from the conclusive conference organized at Bielefeld in December 2017. The contributions consist of original research papers: they mirror the sc...
Article
Full-text available
Ground-state eigenfunctions of Schrödinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs—differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices—a new class of vertex conditions introduced in the paper. It is shown tha...
Preprint
We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide whenever the graph is bipartite; this leads to a precise relation between their eigenvalues enumerated with multip...
Article
Full-text available
Schrödinger operators on metric graphs with delta couplings at the vertices are studied. We discuss which potential and which distribution of delta couplings on a given graph maximise the ground state energy, provided the integral of the potential and the sum of strengths of the delta couplings are fixed. It appears that the optimal potential if it...
Preprint
Discrete-state stochastic models are a popular approach to describe the inherent stochasticity of gene expression in single cells. The analysis of such models is hindered by the fact that the underlying discrete state space is extremely large. Therefore hybrid models, in which protein counts are replaced by average protein concentrations, have beco...
Article
In this paper self-adjoint realizations of the formal expression Aα:=A+α⟨ϕ,⋅⟩ϕ are described, where α∈R∪{∞}, the operator A is self-adjoint in a Hilbert space H and ϕ is a supersingular element from the scale space H−n−2(A)∖H−n−1(A) for n⩾1. The crucial point is that the spectrum of A may consist of the whole real line. We construct two models to d...
Article
Schrödinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of the Ambartsumian theorem is proven under the assumption that the vertex conditions are asymptotically properly connecting and asymptotically standard. By...
Article
Full-text available
A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models is extremely challenging due to their large discrete state space. We therefore propose a hybrid approximation...
Article
We prove that the upper spectral estimate for quantum graphs due to Berkolaiko–Kennedy–Kurasov–Mugnolo [5] is sharp.
Preprint
Full-text available
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or localised nature to a graph impact the spectrum of the Laplacian. Many of these principles are entirely new, these...
Article
Full-text available
In this paper we study Schrödinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending only on the graph and the potential—on the difference between the \(n^\mathrm{th}\) eigenvalues of the Laplace and Schrödinger operators are obtained. This in turn allows us to prove an extension of the classical Ambart...
Chapter
I got to know Mikael in the eighties, when I was a PhD student at Stockholm University. I had completed a number of graduate courses and became interested in complex analysis. Mikael was a young lecturer in Stockholm, and our overlapping interest in that subject brought us into each other’s orbits.
Chapter
1959-01-01. Kjell Alrik Mikael Pettersson is born in Västerås, Sweden. Mother: Britt Gunvor Emilia Pettersson, later with the family name Elfström. Father: Werner Siems. Stepfathers: Kjell Pettersson and Hans Elfström.
Article
Full-text available
We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds is the well-known inequality of Fiedler, of which we give a n...
Article
Full-text available
Not necessarily self-adjoint quantum graphs -- differential operators on metric graphs -- are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator is $ \mathcal P \mathcal T$-symmetric, then its spectrum has reflection symmetry with respect to the real line...
Article
Full-text available
The paper presents a new model of the visual-motor monitoring which features two control loops - objective and subjective ones. This model allows to explain the phenomenon of cognitive dissonance that is experienced by operators in solving sophisticated problems. The results of experimental studies comprising the control of an object under conditio...
Book
This book is dedicated to the memory of Mikael Passare, an outstanding Swedish mathematician who devoted his life to developing the theory of analytic functions in several complex variables and exploring geometric ideas first-hand. It includes several papers describing Mikael’s life as well as his contributions to mathematics, written by friends of...
Article
Equi-transmitting scattering matrices are studied. A complete description of such matrices up to order five is given. It is shown that the standard matching conditions matrix is essentially the only equi-transmitting matrix for orders 3 and 5. For orders 4 and 6, there exists other equi-transmitting ones but all such matrices have zero trace.
Conference Paper
Full-text available
The purpose of the paper is to develop methodological bases for assessing vocational aptitude of human-operators of man-machine systems. The model of vocational aptitude and the process of decision-making in the class of hierarchical systems were developed based on the hierarchy analysis method. The experimental research of vocational aptitude asse...
Article
Full-text available
How ideas of P T -symmetric quantum mechanics can be applied to quantum graphs is analyzed, in particular to the star graph. The class of rotationally symmetric vertex conditions is analyzed. It is shown that all such conditions can effectively be described by circulant matrices: real in the case of odd number of edges and complex having particul...
Article
Spectral properties of the Schrödinger operator on a finite compact metric graph with delta-type vertex conditions are discussed. Explicit estimates for the lowest eigenvalue (ground state) are obtained using two different methods: Eulerian cycle and symmetrization techniques.
Article
Full-text available
In this paper vertex conditions for the differential operator of fourth derivative on the simplest metric graph – the Y-graph, – are discussed. In order to make the operator symmetric one needs to impose extra conditions on the limit values of functions and their derivatives at the central vertex. It is shown that such conditions corresponding to t...
Article
Full-text available
We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We invest...
Chapter
Lower and upper estimates for the spectral of the Laplacian on a compact metric graph are discussed. New upper estimates are presented and existing lower estimates are reviewed. The accuracy of these estimates is checked in the case of complete (not necessarily regular) graph with large number of vertices.
Article
We study the spectral gap, i.e. the distance between the two lowest eigenvalues for Laplace operators on metric graphs. A universal lower estimate for the spectral gap is proven and it is shown that it is attained if the graph is formed by just one interval. Uniqueness of the minimizer allows to prove a geometric version of the Ambartsumian theorem...
Article
Full-text available
Reflectionless equi-transmitting unitary matrices are studied in connection to matching conditions in quantum graphs. All possible such matrices of size 6 are described explicitly. It is shown that such matrices form 30 six-parameter families intersected along 12 five-parameter families closely connected to conference matrices.
Article
Full-text available
We discuss lower and upper estimates for the spectral gap of the Laplace operator on a finite compact connected metric graph. It is shown that the best lower estimate is given by the spectral gap for the interval with the same total length as the original graph. An explicit upper estimate is given by generalizing Cheeger's approach developed origin...
Article
The inverse problem for the magnetic Schrödinger operator on the lasso graph with different matching conditions at the vertex is investigated. It is proven that the Titchmarsh-Weyl function known for different values of the magnetic flux through the cycle determines the unique potential on the loop, provided the entries of the vertex scattering mat...
Article
The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.
Article
Full-text available
The spectral gap for Laplace operators on metric graphs is investigated in relation to graph's connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap is not one-to-one. The size of the spectral gap depends not...
Book
The conference Operator Theory, Analysis and Mathematical Physics – OTAMP is a regular biennial event devoted to mathematical problems on the border between analysis and mathematical physics. The current volume presents articles written by participants, mostly invited speakers, and is devoted to problems at the forefront of modern mathematical phys...
Article
Full-text available
Schrödinger operators on metric trees are considered. It is proven that for certain matching conditions the Titchmarsh-Weyl matrix function does not determine the underlying metric tree, i.e. there exist quantum trees with equal Titchmarsh-Weyl functions. The constructed trees form one-parameter families of isospectral and isoscattering graphs.
Article
An explicitly solvable model of the gated Aharonov–Bohm ring touching a quantum wire is constructed and investigated. The inverse spectral and scattering problems are discussed. It is shown that the Titchmarsh–Weyl matrix function associated with the boundary vertices determines a unique electric potential on the graph even though the graph contain...
Article
Full-text available
An introduction into the area of inverse problems for the Schrödinger operators on metric graphs is given. The case of metric nite trees is treated in detail with the focus on matching conditions. For graphs with loops we show that for almost all matching conditions the potential on the loop is not determined uniquely by the TitchmarshWeyl function...
Article
Full-text available
In this article an operator theoretic interpretation of the generalized Titchmarsh-Weyl coefficient for the Hydrogen atom differential expression is given. As a consequence we obtain a new expansion theorem in terms of singular generalized eigenfunctions. KeywordsTitchmarsh-Weyl coefficient–Singular differential operator–Generalized Nevanlinna fun...
Article
Full-text available
Schroedinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner-von Neumann type is considered. The asymptotics of generalized eigenvectors for the values of the spectral parameter from the upper half-plane and on the absolutely continuous spectrum is established. Weyl-Titchmarsh type formula for...
Article
Full-text available
The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique...
Article
Full-text available
The inverse problem for Schrodinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh-Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to recon- stru...
Article
Full-text available
Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of...
Article
Full-text available
Quantum graphs having one cycle are considered. It is shown that if the cycle contains at least three vertices, then the potential on the graph can be uniquely reconstructed from the corresponding Titchmarsh–Weyl function (Dirichlet-to-Neumann map) associated with graph's boundary, provided certain non-resonant conditions are satisfied.
Article
Full-text available
A new type of point interactions for the Laplacian in \mathbb R3 { \mathbb R^3 } is constructed generalizing classical Fermi pseudopotential. This model leads to a new resolvent formula and a non-trivial scattering matrix in p-channel.
Article
Full-text available
The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions...
Article
Full-text available
Laplace operators on metric graphs are considered. It is proven that for compact graphs the spectrum of the Laplace operator determines the total length, the number of connected components, and the Euler characteristic. For a class of non-compact graphs the same characteristics are determined by the scattering data consisting of the scattering matr...
Article
Full-text available
Three different inverse problems for the Schrödinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator (dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary cont...
Article
The inverse spectral problem for Schrödinger operators on finite compact metric graphs is investigated. The relations between the spectral asymptotics and geometric properties of the underlying graph are studied. It is proven that the Euler characteristic of the graph can be calculated from the spectrum of the Schrödinger operator in the case of es...
Article
Full-text available
The essential spectrum of the singular matrix differential operator of mixed order determined by the operator matrix (Equation Presented) is studied. Investigation of the essential spectrum of the corresponding self-adjoint operator is continued but now without assuming that the quasi-regularity conditions are satisfied. New conditions that guarant...
Article
Full-text available
We investigate the spectral properties of Schrödinger operators with point interactions, focusing attention on the interplay between level repulsion (von Neumann-Wigner theorem) and the symmetry of the confi.guration of point interactions. The explicit solution of the problem allows observing level repulsion for two centers. For a large number of c...
Article
Full-text available
Wigner–von Neumann type perturbations of the periodic one-dimensional Schrödinger operator are considered. The asymptotics of the solution to the generalized eigenfunction equation is investigated. It is proven that a subordinated solution and therefore an embedded eigenvalue may occur at the points of the absolutely continuous spectrum satisfying...
Book
This volume contains lectures delivered by the participants of the international conference "Operator Theory, Analysis and Mathematical Physics" (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland on July 6-11, 2004. The idea behind these lectures was to present interesting ramifications of operator...
Chapter
Full-text available